5.2b Change of Base of Logarithms

Example 1:
Find the value of the following:
a. ${\log }_{25}100$
b. ${\log }_{3}0.45$

Answer:
$\begin{array}{l}\text{(a) }{\log }_{25}100=\frac{{\log }_{10}100}{{\log }_{10}25}=\frac{{\log }_{10}{10}^2}{{\log }_{10}25}=\frac{2}{1.3979}=1.431\\ \\ \\ \text{(b) }{\log }_{3}0.45=\frac{{\log }_{10}0.45}{{\log }_{10}3}=\frac{-0.3468}{0.4771}=-0.727\end{array}$

Example 2
Find the value of the following.
(a) ${\log }_{4}8$
(b) ${\log }_{125}5$
(c) ${\log }_{81}27$
(d) ${\log }_{16}64$